A Comprehensive Review of Fixed Point Theorems on Various Metric Spaces and Their Applications

Aronszajn and Panitchpakdi developed hyperconvex metric spaces to expand Hahn-theorem Banach's beyond the real line to more generic spaces. The aim of this short article is to collect and combine basic notions and results in the fixed point theory in the context of hyperconvex metric spaces. In this paper, we first introduce the definitions of hyperconvex metric spaces, nonexpansive retract, externally hyperconvex and bounded subsets, and admissible subsets. We shall review and explore some fundamental characteristics of hyperconvexity. Next, we introduce the Knaster–Kuratowski and Mazurkiewicz (KKM) theory in hyperconvex metric spaces and related results. Furthermore, we find the relationship between extremal points and hyperconvexity and related properties. Furthermore, we have highlighted some known consequences of our main results. Fixed point theorems are fundamental tools in mathematical analysis, and have been used for diverse purposes including optimization, differential equations and dynamical systems. Fixed point theory was initiated in the case of standard metric spaces and subsequently expanded to be-metrical, convex be-metallic etc., as we needed more generalised conditions for broader range of maps from a wider variety of intuitive settings. This paper investigates fixed point theorems in convex b-metric spaces, which studies by Chen et al. This study aimed to synthesize and interpret the results in terms of practical implications as well as theoretical contributions that emerge from these findings, which are discussed later on this discussion section. In situations where traditional metric constraints are restrictive, fixed point theorems can be implemented in a class of spaces using convex b-metric structure by each relaxing and combining part of triangle inequality with the condition involved from standard assumption about Banach contraction. The results in this paper are existence and uniqueness theorems, common fixed point theorem of two mappings under some contractive conditions, a common coupled coincidence point result for four self maps, thereby indicating that convex b-metric spaces can be used but desired to solve boundary value problems (BVP), stabilization of dynamic systems over bounded closed sets with Lyapunov functions. Applications show how these abstract sittings could play a more elegant role in nonlinear optimization too by identifying optimal solutions. We also compare our result of convex b-metric spaces with other generalized metrics and we discuss their advantages for future research. In particular the above review sheds light on unlimitedness of these strikingly handy class, convex b-metric spaces alongside its implication for rich maths art instead.